Validation of the Axial Thrust Estimation Method for Radial Turbomachines

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Tiainen Jonna, Jaatinen-Värri Ahti, Grönman Aki, Sallinen Petri, Honkatukia Juha, Hartikainen Toni

Tiainen, J. et al. (2021) Validation of the Axial Thrust Estimation Method for Radial

Turbomachines. International Journal of Rotating Machinery, Volume 2021, Article ID 6669193, p. 18. DOI: 10.1155/2021/6669193

Publisher's version Hindawi

International Journal of Rotating Machinery

10.1155/2021/6669193

© 2021 Jonna Tiainen et al.

Research Article

Validation of the Axial Thrust Estimation Method for Radial Turbomachines

Jonna Tiainen , ¹ Ahti Jaatinen-Värri , ¹ Aki Grönman , ¹ Petri Sallinen , ¹ Juha Honkatukia , ¹ and Toni Hartikainen ²

1

Lappeenranta-Lahti University of Technology LUT, Lappeenranta, Finland

2

Aurelia Turbines Oy, Lappeenranta, Finland

Correspondence should be addressed to Jonna Tiainen; jonna.tiainen@lut. fi Received 18 November 2020; Accepted 3 February 2021; Published 24 February 2021 Academic Editor: Cheng Xu

Copyright © 2021 Jonna Tiainen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The fast preliminary design and safe operation of turbomachines require a simple and accurate prediction of axial thrust. An underestimation of these forces may result in undersized bearings that can easily overload and su ffer damage. While large safety margins are used in bearing design to avoid overloading, this leads to costly oversizing. In this study, the accuracy of currently available axial thrust estimation methods is analyzed by comparing them to each other and to theoretical pressure distribution, numerical simulations, and new experimental data. Available methods tend to underestimate the maximum axial thrust and require data that are unavailable during the preliminary design of turbomachines. This paper presents a new, simple axial thrust estimation method that requires only a few preliminary design parameters as the input data and combines the advantages of previously published methods, resulting in a more accurate axial thrust estimation. The method is validated against previously public data from a radial pump and new experimental data from a centrifugal compressor, the latter measured at Lappeenranta- Lahti University of Technology LUT, Finland, and two gas turbines measured at Aurelia Turbines Oy, Finland. The maximum deviation between the estimated axial thrust using the hybrid method and the measured one is less than 13%, while the other methods deviate by tens of percent.

1. Introduction

The design of the bearings and sealing of turbomachines is based on the maximum axial thrust occurring during opera- tion. Its maximum value is estimated at the preliminary tur- bomachinery design stage, and the importance of an accurate estimation is highlighted in, for example [1]. A simple and accurate method for axial thrust estimation is of paramount importance for the fast preliminary design and safe operation of the machine since an underestimation would result in undersized bearings, which can lead to overloading and even damage the bearings. To prevent undersizing, safety margins as large as two times the maximum axial thrust are used [2].

It is envisaged that having an accurately estimated axial thrust would enable the use of lower safety margins in the bearing design and thus prevent costly oversizing.

The axial thrust can be estimated by applying Newton’s second law, which requires the reliable prediction of pressure distributions on the shroud and back disk sides of an impel- ler. According to the numerical results of the flow fields inside the impeller back disk cavity [3], the average circum- ferential velocity in the cavity is approximately half the impeller’s local circumferential velocity, which is the usual assumption in axial thrust estimation methods [4, 5]. The assumption of a constant swirl factor of 0.5 simplifies the axial thrust estimation. However, the factor changes when the compressor operates at o ff-design conditions, the factor is slightly higher in the front side cavity (of the shrouded impeller, on average 0.6 in [6]) than in the back disk cavity (on average 0.4 in [6]), and this difference increases at the off-design conditions [6]. As the swirl factor depends on, e.g., the fluid properties, geometry of the cavity, and

Volume 2021, Article ID 6669193, 18 pages

https://doi.org/10.1155/2021/6669193

rotational speed, its value cannot be analytically estimated, but computational fluid dynamics (CFD) simulation is required instead [7].

The static pressure distribution in the radial direction on the back disk side varies depending on whether the leakage flow rate through the cavity is negative, positive, or zero [3]. A zero leakage flow rate can be related to, e.g., turbo- chargers, positive flow (flow direction towards the impeller outlet) to multistage and vacuum compressors, and negative flow (away from the impeller outlet) to compressors deliver- ing at overpressure and working in an open cycle. The modeled shapes of the static pressure distributions in the centrifugal compressor [3] at different leakage flow rates were similar to those measured in the radial pump [8].

The impeller pressure distribution can be estimated using numerical models; however, accounting for the back disk cavity with a possible radial labyrinth seal complicates the model. Besides, simpler tools than CFD are generally pre- ferred at the preliminary design stage. Optionally, analytical or semiempirical expressions can be used to estimate the pressure distribution. Already in 1955, a theoretical equation [9] was presented for the pressure distribution in a labyrinth seal in cases of negative and positive flows. Later, a method [5] was proposed assuming that a fluid element is in radial equilibrium so that the pressure forces balance the centrifu- gal forces and pressure varies as a function of the impeller radius. The second method [4] approximates the pressure distribution by assuming a relation between velocity and pressure in terms of total relative pressure. The third method [10] requires data on the leakage flow rate through the back disk cavity to estimate the pressure distribution. However, there is no generally accepted formulation for the leakage flow rate in the literature. The fourth method [11] proposed for turbochargers obtains the outlet pressure of the compres- sor ’s impeller and the inlet pressure of the turbine’s rotor from the reaction degree, and it assumes constant pressure on the back disk side, since in turbochargers, there is typically no leakage flow through the back disk cavity.

This study compares the first [5], second [4], and fourth [11] methods, which are hereafter named after their inven- tors as the methods of Larjola, Japikse, and Nguyen-Schäfer, respectively. These methods are compared to each other as well as to new experimental data, numerical simulations, and the theoretical equation [9], hereafter named after Kear- ton. Section 2 describes the experimental methods for the centrifugal compressor with the new data on axial thrust at different operating points measured at the Lappeenranta- Lahti University of Technology LUT. The compressor is used as the example case in the comparison of different axial thrust estimation methods, which are presented in Section 3. Section 4 describes the numerical methods, and Section 5 compares the analytical, experimental, and numerical results.

The comparison provides valuable information about the accuracy of the axial thrust estimation methods. To estimate the axial thrust more accurately than the currently available methods, a new method is proposed, which is presented in Section 6. It is validated against both previously public and new experimental data on the centrifugal compressor, radial pump, and two gas turbines. The validation cases are

presented in Section 7, and the validation results are dis- cussed in Section 8. The last section concludes this study.

The novelties of this study are the presented hybrid method, its validation against different radial turbomachines, and new experimental data on the axial thrust of the centrif- ugal compressor. The advantages of the hybrid method are its better accuracy compared to the currently available methods and its simplicity, since it only requires data that are available at the preliminary design. The hybrid method has scienti fic implications for designers, engineers, and scientists working with turbomachinery: firstly, it can strengthen the design process with a simple and more accurate axial thrust estima- tion, which ensures that the operation of a turbomachine as axial thrust is not underestimated; secondly, it can speed up the design process as numerical modeling with the back disk cavity is not necessary.

2. Experimental Methods

The closed loop test rig in the Laboratory of Fluid Dynamics at LUT University, Finland, was used to measure the axial thrust of a centrifugal compressor. The studied compressor was designed at LUT. It had an unshrouded impeller, nine full and nine splitter blades, a parallel wall vaneless diffuser, and a volute. The shaft of the compressor was positioned ver- tically, and there was a radial labyrinth seal in the back disk cavity. The compressor was controlled with active magnetic bearings. A schematic view of the axial bearings and force components of the net axial thrust is presented in Figure 1.

The following parameters were measured: static pressure and temperature at the compressor stage inlet and outlet;

total pressure, total temperature, and static pressure at the impeller outlet; static pressure at the diffuser outlet; and ambient pressure, mass flow rate, and axial thrust. The per- formance measurement setup complied with ISO 5389. The measurements of nine operation points took approximately 2.5 hours as the steady state of one point was achieved within 15 minutes. Total pressure and temperature at the impeller outlet (r = 145:50 mm = 1:07r 2 ) were measured with a Kiel probe. Static pressure at the impeller outlet was measured at the radius of 153.50 mm (r = 1:13r 2 ), and static pressure at the diffuser outlet was measured at the radius of 264.00 mm (r = 0:97r 3 ). The measurement locations are shown in Figure 2. The axial clearance between the impeller back disk and the casing on the back side was approximately 5 mm.

The maximum measurement uncertainties were 46 N for axial thrust, 2.19 kPa for diffuser outlet pressure, and 0.86 kPa for impeller outlet pressure. The axial thrust was measured with an active magnetic bearing control software, which was calibrated with the load cell. The mass of the shaft and the axial thrust of the fan, which cooled down the electric motor, were accounted for in the net axial thrust calculations.

The effect of the fan was tested separately, without the compressor impeller. The axial thrust of the fan was approx- imately 15% of the compressor ’s axial thrust at the compres- sor ’s design point.

The e ffect of thermal expansion on the axial thrust mea-

surement was tested before and after the 2.5-hour test period.

Before the test period, the measured axial thrust of the cold nonrotating compressor was 26 N, and after the test period, the axial thrust of the nonrotating compressor was -49 N;

the difference between these values was 75 N. As the mea- surement uncertainty of the axial thrust was 46 N and the effect of thermal expansion (75 N) was less than 6% of the axial thrust measured during the compressor operation (1.3 –2.2 kN), it can be concluded that the effect of thermal expansion was negligible in this study. The measured operat- ing points are presented in Table 1 and in the compressor performance map in Figure 3.

3. Analytical Methods

Di fferent methods have been presented for estimating axial thrust by applying Newton ’s second law. Below, the methods proposed by Nguyen-Schäfer [11], Japikse [4], and Larjola [5] are presented. In addition, the theoretical prediction for the pressure distribution in a labyrinth seal proposed by Kearton [9] is shown.

3.1. Nguyen-Schäfer. Nguyen-Schäfer ’s method [11] for the calculation of the axial thrust load on an automotive turbo- charger assumes steady state flow and negligibly low viscous friction at the walls. The axial thrust on the compressor side of the turbocharger consists of the pressure force at the com-

pressor inlet F inlet , the pressure force at the shroud F shroud , the impulse force F impulse , and the pressure force at the back disk F back disk , as follows:

F = F inlet + F shroud + F impulse − F back disk , ð1Þ

F inlet = p ₁ π

4 d ² _1,s , ð2Þ

F shroud = p ₁ + p ₂ 2

 π

4  d ² 2 − d ² _1,s 

, ð3Þ

F impulse = q ² _m R air T 1

p 1 π/4 d  ² _1,s − d ² _1,h  , ð4Þ F back disk = p 2

π

4 d ² ₂ − d ² 0

 

, ð5Þ

where the terms d, p, q _m , R, and T refer to the diameter, pres- sure, mass flow rate, specific gas constant, and temperature, respectively. The subscripts 0, 1, 2, h, and s refer to the shaft, compressor inlet, impeller outlet, hub, and shroud, respec- tively. The pressure forces are integrated over the surface, and the impulse force is solved from the momentum theo- rem. The equations are analogous with the turbine. The method requires the parameters presented in Table 2.

Gravity Bearing forces Axial thrust

of the fan

Pressure forces on the back disk

Pressure forces on the shroud and compressor inlet

Impulse forces

Figure 1: Schematic view of the studied centrifugal compressor, the locations of the axial magnetic bearings, and the force components of the net axial thrust.

r diffuser outlet pressure = 264.00 mm r impeller outlet pressure = 153.50 mm

r Kiel probe = 145.50 mm

G

DETAIL G

Figure 2: Measurement locations of total pressure and temperature (Kiel probe) at the impeller outlet and static pressure at the impeller and

di ffuser outlets. Detail G shows the dimensions of the static pressure taps.

The pressure at the impeller outlet is estimated using the reaction degree.

p ₂ = p ₁ 1 + RD p ₅ p 1

  _{ð γ−1 Þ/γ}

− 1

" #

( ) _{γ/ γ−1 ð Þ}

, ð6Þ

where subscript 5 refers to the compressor stage outlet, the term γ refers to the ratio of specific heats, and the reaction degree RD is de fined as follows:

RD = Δh impeller

Δh stage

, ð7Þ

where the term h refers to the specific enthalpy. At the back disk surface, constant pressure is assumed. The assumption is applicable in the case of turbochargers, where there are often no leakage flows into or out of the back disk cavity [12].

The discrepancy between the axial thrust estimated using Nguyen-Schäfer’s method and the axial thrust esti- mated using CFD was less than 10% for an automotive turbocharger [11].

3.2. Japikse. Under Japikse ’s [4] method, the cavity pressure distribution is approximated by assuming a relation between velocity and pressure in terms of total relative pressure as follows:

p _t = p 2 + w ² ₂ 2 − U ² ₂

2

 

ρ 2 = p 1 ′ + w ¹ ′ ² 2 − U ₁ ′ ²

2

!

ρ 1 ′, ð8Þ

where the terms U, w, and ρ refer to the circumferential velocity, relative velocity, and density, respectively. The term p ₁ ′ refers to the pressure just ahead of the seal on the shroud side, and it is assumed that the relative velocity in the cavity is some fraction of the local rotor circumferential velocity

w = f U: ð9Þ

The fraction is generally simplified to be constant in the range of 0.5 to 1.0, but it can be expressed as a function of radius. In this study, a constant fraction f of 0.5 is used.

The axial thrust consists of the pressure force at the impeller nose F nose , the pressure force at the impeller eye F eye , the pressure force at the shroud F shroud , the impulse force F impulse , and the pressure force at the back disk F back disk as follows:

F = F nose + F eye + F shroud + F impulse − F back disk , ð10Þ

F nose = p _1,h π

4 d ² _1,h , ð11Þ

F eye = p _1,h + p _1,s 2

  π

4 d ² _1,s − d ² 1,h

 

, ð12Þ

F shroud = 1 2

ρ

2  1 − f ² 

ð 2πn Þ ² d 2

2

  2

+ d _1,s 2

  2

" #

(

+ p 1 ′ − ρ ¹ ′

2  1 − f ²  U 1 ′ ²

)

· π d 2

2

  2

− d _1,s 2

  2

" #

, ð13Þ Table 1: Measured operating points.

Point n

(Hz) q _m (kg/s)

p ₁ (kPa)

T 1

(

^°

C) p 3

(kPa)

p amb

(kPa)

9 462 1.57 94.5 23.8 226.2 100.2

8 461 1.77 94.1 24.6 216.9 100.2

7 461 2.06 92.7 25.3 201.5 100.2

6 393 1.31 96.9 20.3 184.3 100.2

5 392 1.53 95.7 20.8 176.1 100.3

4 392 1.76 94.7 21.5 165.3 100.3

3 323 1.06 98.0 18.4 152.5 100.3

2 323 1.25 97.3 18.9 146.8 100.3

1 322 1.45 97.3 18.6 140.0 100.3

0.5 3.0

1.0 462 Hz 416 364 76 79

79 76

81 323 82

1.5

Reference mass flow rate (kg/s)

T o ta l-t o-t o tal p ressur e ra tio [–]

2.0 2.5

2.5

2.0

1.5

1.0

Figure 3: Compressor performance map with the measured operating points.

Table 2: Parameters of the studied centrifugal compressor at the point of maximum axial thrust (point 9) used in the method proposed by Nguyen-Schäfer.

Pressure at the impeller inlet p ₁ 94.5 kPa Temperature at the impeller inlet T 1 297 K Pressure at the compressor outlet p ₅ 231.2 kPa Impeller inlet hub diameter d _1,h 40.5 mm Impeller inlet shroud diameter d _1,s 134.9 mm

Impeller outlet diameter d ₂ 270.9 mm

Shaft diameter d 0 32.0 mm

Speci fic enthalpy increase, impeller Δh imp 116.4 kJ/kg Speci fic enthalpy increase, stage Δh st 150.7 kJ/kg

Ratio of speci fic heats γ 1.4

Speci fic gas constant R air 287 J/kg K

Mass flow rate q _m 1.57 kg/s

F impulse = q _m c 1 , ð14Þ

F back disk = 1 2

ρ

2  1 − f ² 

ð 2πn Þ ² d 2

2

  2

+ d 0

2

  2

" #

(

+ p 1 ′ − ρ ¹ ′

2  1 − f ²  U 0 ′ ²

)

· π d 2

2

  2

− d 0

2

  2

" #

, ð15Þ where the terms c, n, and ρ refer to the absolute velocity, rotational speed, and average gas density in the cavity, respectively.

The method requires the parameters shown in Table 3.

The impeller inlet absolute velocity c 1 in (14) can be solved from the mass flow rate:

c 1 = q _m

ρ ₁ π/4 d  ² _1,s − d ² _1,h  : ð16Þ Japikse [4] does not give any estimation for the pressures just ahead of the seal on either the shroud side p 1 ′ or the back disk side p ₀ ′. However, in this paper, the impeller is unshrouded and has no seal on the shroud side. Therefore, the pressure at the impeller inlet p ₁ is used as the pressure p ₁ ′ on the shroud side, and the ambient pressure p _amb is used as the pressure ahead of the seal on the back disk side.

Under Japikse’s method [4], the axial thrust is calculated using the average pressure in the shroud and back disk cavi- ties, even though the pressure distribution could be written as a function of radius.

3.3. Larjola. The method proposed by Larjola [5] di ffers from the others in that the pressure distribution is written as a function of radius. Therefore, the pressure distribution becomes more accurate than when the pressure is approxi- mated to change linearly in the cavity. The axial thrust con- sists of the pressure force on the shroud side F shroud , the impulse force F impulse , and the pressure force on the back disk side F back disk as follows:

F = F shroud + F impulse − F back disk : ð17Þ

To calculate the pressure distribution, it is assumed that the fluid element is in radial equilibrium so that the pressure forces balance the centrifugal forces.

1 ρ

dp dr = c ² _u

r , ð18Þ

c _u = f ωr, ð19Þ

⇒dp = ρ f ω ð Þ ² rdr, ð20Þ where the factor f accounts for the slip, and the terms c _u , r, and ω are the absolute tangential velocity, radius, and angular

velocity, respectively. The equation for pressure at an arbi- trary point is derived when the isentropic flow equation

ρ ρ ₂ = p

p 2

  1/ γ

ð21Þ

is combined with the pressure distribution (20).

The parameters needed for the method are listed in Table 4. To calculate the pressure distribution along the back disk, a su fficient number of points between the shaft radius r 0

(or the labyrinth seal radius r _L in the case of the labyrinth seal) and the impeller outlet radius r 2 are chosen. The pres- sure for every point is calculated as follows:

p = p ₂ γ − 1 2 γp 2

ρ ₂ f ² ω ² r ² − r ² 2

 

+ 1

 _{ð γ/γ−1 Þ}

, ð22Þ

where the fraction f equals 0.5 on the back disk side. In the studied compressor, the pressure at the shaft radius r 0 equals the ambient pressure. In this study, it is assumed that the pressure reduces to ambient pressure at the end of the laby- rinth seal (at the radius of 62.5 mm). At the radius of r 2 , the pressure equals p ₂ .

Also on the shroud side, a sufficient number of points between the shroud radius r _1,s and the impeller outlet radius r 2 are chosen to calculate the pressure distribution by apply- ing (22). At the inlet of the radial part r _1,s , the pressure equals p 1 ′ and at the impeller outlet r 2 , the pressure equals p 2 . The fraction f is unknown on the shroud side, thus requiring an initial guess, and it is iterated so that the pressure at the Table 3: Parameters of the studied centrifugal compressor at the point of maximum axial thrust (point 9) used in the method proposed by Japikse.

Pressure at the impeller inlet, hub p _1,h 94.5 kPa Pressure at the impeller inlet, shroud p _1,s 94.5 kPa Pressure at the impeller inlet p ₁ 94.5 kPa Temperature at the impeller inlet T 1 297 K Pressure (just ahead of the seal), shroud p 1 ′ 94.5 kPa Pressure at the impeller outlet p ₂ 199.4 kPa Temperature at the impeller outlet T 2 412 K Pressure just ahead of the seal, back disk p 1 ′ 100.2 kPa Impeller inlet hub diameter d _1,h 40.5 mm Impeller inlet shroud diameter d _1,s 134.9 mm

Impeller outlet diameter d 2 270.9 mm

Shaft diameter d ₀ 32.0 mm

Mass flow rate q _m 1.57 kg/s

Rotational speed n 462 Hz

Fraction f 0.5

impeller shroud radius p 1 ′ from (22) equals the pressure p 1 ′ from (23).

p 1 ′ = p 1 + 1

2 ρ 1 w ² _1,s − ρ 1 ′w ² 2



, ð23Þ

ρ 1 ′ = f p
1 , T 1 ′ 

, ð24Þ

T 1 ′ = f p
1 , h 1s ′ 

, ð25Þ

h 1s ′ = h 1 + 1

2 w ² _1,s − w ² 2

 

η _s,1−2 , ð26Þ

η _s,1−2 = h _2s − h 1

h 2 − h 1

, ð27Þ

where the terms h _1s and h _2s are the isentropic speci fic enthalpies at the impeller inlet and outlet, and η _s,1−2 is the isentropic efficiency. The relative velocities at the impeller

inlet w _1,s and outlet w 2 are solved from the velocity triangles and isentropic flow equation as follows:

w _1,s =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ² 1 + U ² _1,s q

, ð28Þ

U _1,s = πnd 1,s , ð29Þ

w 2 =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ² _r2 + w ² _u2 q

, ð30Þ

c _r2 = q _m

ρ 2 πd 2 b 2 ð 1 − B Þ , ð31Þ

w _u2 = U 2 − c _u2 , ð32Þ

U 2 = πnd 2 , ð33Þ

c _u2 =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ² 2 − c ² r2

q

, ð34Þ

c 2 = Ma ₂ a 2 ð35Þ

Ma ₂ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

γ − 1 p _t2

p ₂

  _{ð γ−1/γ Þ}

− 1

" #

v u

u t , ð36Þ

where the terms a, b, B, Ma, and p _t refer to the sound speed, blade height, boundary layer blockage factor, Mach number, and total pressure, respectively. The subscripts r and u refer to radial and circumferential, respectively.

The pressure forces F shroud and F back disk are integrated at every point between the radius of 0 and r 2 on the shroud and back disk sides:

F = ð r

2

0

2πprdr, ð37Þ

and the impulse force is solved from the momentum theo- rem:

F impulse = q _m c 1 : ð38Þ

The net axial thrust is calculated using (17).

3.4. Kearton. The pressure distribution in the labyrinth seal depends on the flow direction. Kearton [9] presented a the- ory for the pressure distribution in the labyrinth seal with negative flow (out of the cavity). The theory does not account for any possible e ffects of rotation on the pressure distribu- tion and assumes uniform axial clearance. The pressure between the seal rings is de fined as follows:

p _m =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ² 0 − R air T 0 ψ _m

F

q _m,leak A 1

  ₂

s

, ð39Þ

where p 0 is the pressure at the seal inlet, R air is the specific gas constant, T 0 is the temperature at the seal inlet, ψ _m is the area function, F is the coefficient being a function of the pressure ratio in a single constriction, q _m,leak is the theoretical leakage Table 4: Parameters of the studied centrifugal compressor at the

point of maximum axial thrust (point 9) used in the method proposed by Larjola.

Pressure at the impeller inlet p ₁ 94.5 kPa Temperature at the impeller inlet T ₁ 297 K Total pressure at the impeller outlet p _t2 263.9 kPa Pressure at the impeller outlet p 2 199.4 kPa Temperature at the impeller outlet T 2 412 K

Ambient pressure p _amb 100.2 kPa

Impeller inlet hub diameter d _1,h 40.5 mm Impeller inlet shroud diameter d _1,s 134.9 mm

Impeller outlet diameter d ₂ 270.9 mm

Impeller outlet passage height b ₂ 12.2 mm

Shaft diameter d 0 32.0 mm

Ratio of speci fic heats γ 1.4

Speci fic gas constant R air 287 J/kg K

Mass flow rate q _m 1.57 kg/s

Rotational speed n 462 Hz

Boundary layer blockage factor B 0.1

Fraction, back disk f 0.5

Table 5: Values of the coefficient F in Kearton’s theory [9].

p ₁ /p ₀ F

1.00 1.00000

0.95 0.95116

0.90 0.90425

0.85 0.85872

0.80 0.81400

mass flow rate, and A 1 is the area through the first constric- tion. The area function is de fined as follows:

ψ _m = m + 2am m − 1 ð Þ

2r − a 2m − 1 ð Þ , ð40Þ where m is the number of the seal ring, a is the radial distance between any two adjacent rings, and r is the radius of the out- ermost ring. The coe fficient F is the function of pressure ratio in a single constriction having the values shown in Table 5.

Kearton [9] also presented an equation for the theoretical mass flow rate of the leakage through the labyrinth seal.

q _m = A 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F p  ² 0 − p ² _n  R air T 0 ψ _n s

, ð41Þ

where p _n is the pressure at the seal outlet and ψ _n is the area function de fined as follows:

ψ _n = n + 2an n − 1 ð Þ

2r − a 2n − 1 ð Þ , ð42Þ where n is the total number of the seal rings.

In the studied centrifugal compressor, the radial distance between any two adjacent rings a is 1.5 mm, the axial clear- ance is 2.5 mm, the total number of rings n is 16, and the radius of the outermost ring r is 94.5 mm.

4. Numerical Methods

The axial thrust is estimated based on the pressure distribu- tions obtained from the numerical simulation. The inlet pipe, impeller, vaneless di ffuser, and back disk cavity are modeled with ANSYS CFX 19.2. The boundary conditions are based on experimental data of nine different operating points

(Table 1). A total pressure of 92.7 –98.0 kPa and a tempera- ture of 18.4 –25.3 ^° C with a turbulence intensity of 5% are de fined at the computational domain inlet. At the outlets of the di ffuser and back disk cavity, static pressures of 140.0–

226.2 kPa and 100.2 –100.3 kPa are defined, respectively.

According to the findings in [13], the interface between the impeller and diffuser is located at 1:02r 2 . Because the interac- tion between the impeller and the vaneless diffuser is weak, the frozen rotor approach is used to model the interfaces between the stationary and rotating parts. The k − ω SST model is chosen to model turbulence based on the validation for turbomachinery applications [14] and the recommenda- tion in [15]. In the vaneless di ffuser, a structured mesh is used, and an unstructured mesh is used in the impeller and back disk cavity. On average, the dimensionless wall distance y ⁺ is below unity. The meshes of the computational domains with and without the labyrinth seal are shown in Figure 4.

The method proposed by Larjola can be modified accord- ing to whether the cases neglect or account for the labyrinth seal. Therefore, two cases, neglecting and accounting for the labyrinth seal, respectively, are studied numerically. The compressor geometry without the labyrinth seal is used in the mesh independence study. To study mesh independence, the design point of the compressor is modeled. The compar- ison between three di fferent meshes (fine with 60 million ele- ments, medium with 25 million elements, and coarse with 11 million elements) is shown in Figure 5, which presents the pressure distributions on the shroud and back disk sides. Dif- ferent meshes give similar results, and the pressure distribu- tions match well with the design values. The medium mesh with 25 million elements was chosen for the studies. The gray area in Figure 5 presents the discretization error estimated with the method proposed in [16].

Axial thrust caused by the pressure forces on the shroud and back disk sides is more signi ficant than the axial thrust caused by the impulse force. Therefore, it is more important

(a) (b)

Figure 4: Mesh of the computational domain without (a) and with (b) the labyrinth seal.

that the pressure distribution is correctly predicted by the numerical model, and the underestimation of the compressor mass flow rate by the numerical model does not significantly a ffect the net axial thrust.

Table 6 presents the modeled mass flow rate at the com- pressor inlet when the back disk cavity is modeled without and with the labyrinth seal. Results in Table 6 indicate that the leakage flow rate out of the back disk cavity reduces by only approximately 1 g/s (0.1% of the total mass flow rate) when the labyrinth seal is modeled, compared to when the cavity does not include the labyrinth seal. According to the numerical results, the leakage mass flow rate is in the range

80 300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 260 250

200 150 100 50 0

60 million mesh elements 25 million mesh elements 11 million mesh elements Design

(a)

80 300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 260 250

200 150 100 50 0

60 million mesh elements 25 million mesh elements 11 million mesh elements Design

(b)

Figure 5: Numerical results of pressure distributions on the (a) shroud and (b) back disk sides with three different meshes and comparison with the design values. Gray areas present the discretization error.

Table 6: Comparison of measured and modeled mass flow rate without and with the labyrinth seal.

Without seal With seal

Point Exp.

(kg/s)

CFD (kg/s)

Di fference (%)

CFD (kg/s)

Di fference (%)

9 1.57 1.493 -4.7% 1.492 -4.8%

8 1.77 1.620 -8.5% 1.619 -8.5%

7 2.06 1.828 -11.2% 1.828 -11.2%

6 1.31 1.301 -0.9% 1.299 -1.0%

5 1.53 1.442 -6.1% 1.441 -6.1%

4 1.76 1.648 -6.5% 1.649 -6.4%

3 1.06 1.081 1.5% 1.079 1.3%

2 1.25 1.237 -1.0% 1.238 -1.0%

1 1.45 1.454 0.5% 1.454 0.6%

Table 7: Comparison of measured and modeled total-to-total pressure ratio without and with the labyrinth seal.

Without seal With seal

Point Exp.

(-)

CFD (-)

Di fference (%)

CFD (-)

Di fference (%)

9 2.79 2.73 -2.2% 2.73 -2.2%

8 2.69 2.62 -2.6% 2.62 -2.6%

7 2.54 2.48 -2.3% 2.48 -2.4%

6 2.16 2.11 -2.4% 2.11 -2.4%

5 2.08 2.03 -2.6% 2.03 -2.6%

4 1.98 1.95 -1.8% 1.95 -1.8%

3 1.69 1.67 -1.0% 1.67 -1.0%

2 1.64 1.62 -1.1% 1.62 -1.1%

1 1.57 1.56 -0.6% 1.56 -0.6%

0.0 1.2

0.2 0.4

Pressure/Reference pressure [–]

R adi us/I m p eller o u tlet radi us [–]

0.6 0.8 1.0 1.2

1.0 0.8 0.6 0.4 0.2 0.0

CFD w/o seal (–0.63%) CFD w seal (–0.59%) Sun et al. (–1%) Sun et al. (0%) Gantar et al. (–1%) Gantar et al. (0%)

Figure 6: Numerical results of pressure distribution on the back

disk side without and with the labyrinth seal. Gray areas present

the discretization error, and relative leakage flow rate is shown in

parentheses in the legend. Numerical results are compared with

those presented in [3] and to the experimental results in [8].

of 5 to 10 g/s, which is 0.4–0.7% of the compressor’s mass flow rate. Kearton’s [9] theory predicts that the leakage mass flow rate is ten times larger than the numerical result, namely, 79 –159 g/s (5–10% of the compressor’s mass flow rate). The leakage mass flow rate of the studied compressor is approximately 1 –2% of the compressor’s mass flow rate.

Therefore, the modeled value is closer to the actual one than the value based on Kearton ’s [9] theory. In (41), the pressures at the seal inlet and outlet are, respectively, the measured static pressure at the impeller outlet and the ambient pres- sure. Because the pressures exactly at the seal inlet and outlet are not measured, the use of impeller outlet and ambient pressure values results in an error in the leakage mass flow estimation, as will be demonstrated later.

The di fference between the modeled and measured mass flow rates is in the range of 0.5% (point 1) to 11.2% (point 7).

However, the modeled total-to-total pressure ratio π t1−t2 is much closer to the measured one, the maximum difference being 2.6% (points 5 and 8), as shown in Table 7. In Figure 6, the modeled pressure distribution on the back disk side is compared to the numerical results in the case of a cen- trifugal compressor [3] and to the experimental results in the case of a radial pump [8]. The relative values of the leakage mass flow rates are given in parentheses in the legend. The modeled pressure distributions with and without the laby- rinth seal are similar to those presented in [3], but di fferent cavity geometries cause di fferences.

5. Comparison of the Analytical Methods The axial thrust values and pressure distributions from the abovementioned analytical methods are compared to the measurements and modeled results. The nine measured operating points presented in Table 1 and the data presented in Tables 2–4 are substituted in the equations of the three methods to estimate the axial thrust. First, the net axial thrust estimated using these methods is compared to the measured and modeled values. Second, the pressure distributions esti- mated by the methods are compared to the measured values, the modeled distributions, and the theoretically predicted distributions.

5.1. Net Axial Thrust. When the parameters of the point of maximum axial thrust (point 9 in Table 1 and the data in Tables 2 –4) of the studied compressor are substituted in the abovementioned equations, three axial thrust calculation methods give the results presented in Table 8. The results given by these analytical methods are compared to the mea- sured axial thrust in the same table, and the comparison indi- cates that Larjola’s method gives the best, albeit a still underestimated, prediction.

The estimations for other operating points made with three analytical methods and numerical model are shown in Figure 7. In Figure 7(a), the axial thrust estimations are compared to the measured axial thrust. In Figure 7(b), the relative di fference between the estimations and the measured axial thrust is presented. The results in Figure 7 indicate that Nguyen-Schäfer ’s method gives relatively close estimations for axial thrust at low rotational speeds, but it overestimates

the axial thrust by 49% at the point of maximum measured axial thrust (point 9) when leakage is neglected. When leak- age is accounted for, Nguyen-Schäfer ’s method underesti- mates the axial thrust by 70 –85%. Japikse’s method underestimates the axial thrust by 55 –72%. When the laby- rinth seal is neglected, the greatest underestimation under Larjola’s method is 64% (point 1) and the greatest overesti- mation is 24% (point 9). The most realistic case is Larjola’s method with a labyrinth seal as the estimation in this case deviates from the measured axial thrust by only -12% at the point of maximum axial thrust (point 9). The results of the numerical simulation underestimate the axial thrust by 26 –58%.

The estimation of the net axial thrust can be divided into three parts, namely, the pressure force acting on the shroud side, the pressure force acting on the back disk side, and the impulse force. These three forces, estimated by the three methods and numerical simulations, are presented in Figure 8. As mentioned above, the impulse force does not contribute to the net axial thrust as significantly as the pres- sure forces. Figure 8(a) indicates that all the methods and numerical models give an almost equal estimation of the pressure force on the shroud side. Figure 8(c) indicates that the numerical results of the impulse force di ffer slightly from the estimations of the three methods, which results from the di fference in the modeled mass flow rate (Table 6). The most signi ficant difference in the axial thrust estimation occurs in the pressure force on the back disk side (Figure 8(b)).

5.2. Pressure Distributions. The difference between the axial thrust estimations, especially on the back disk side, results from the estimated pressure distribution as the axial thrust is a product of pressure and area. The pressure distributions on the shroud and back disk sides of the studied compressor are sketched in Figure 9 for the design point (point 8). The pressure distributions on the shroud and back disk sides, with and without the leakage/labyrinth seal, are presented in Figures 10 –12. The figures present two extreme points (operation points 1 and 9, listed in Table 1) since the other points fall within these two extremes. Figure 10 presents the pressure distribution on the shroud side estimated with the three methods and CFD. The measured static pressures at the impeller inlet, impeller outlet, and diffuser outlet are also presented. Figure 11 presents the pressure distribution on the back disk when leakage and the labyrinth seal are neglected.

Table 8: Maximum axial thrust (point 9) estimated with three analytical methods and compared with the measured one. In Nguyen-Schäfer ’s and Larjola’s methods, the leakage through the labyrinth seal can be either neglected or accounted for.

Method Without leakage/seal (kN)

With leakage/seal (kN) Nguyen-

Schäfer 3.27 0.65

Japikse 0.91

Larjola 2.71 1.93

Measurement 2 :19 ± 0:046

Figure 12 presents the pressure distribution on the back disk when leakage and labyrinth seal are accounted for.

Figure 10 indicates that the pressure distribution on the shroud side does not deviate greatly between the di fferent methods and the numerical model. However, the linear assumption of Japikse ’s method with the fraction f of 0.5 gives a better distribution than that predicted by other methods.

Figure 11 clearly shows that the assumption of constant pressure in the back disk cavity is not valid in the studied cen- trifugal compressor, even though it might be valid in the case of turbochargers [11]. The assumption of nearly constant pressure on the back disk surface results in overestimated pressure distribution on the back disk side and axial thrust under Nguyen-Schäfer ’s method. However, if the leakage flow through the back disk cavity is accounted for, as under

1 2 3 4 5 6 7 8 9

Point number 0.0

–0.5 –1.0 –1.5 –2.0 –2.5 –3.0 –3.5

Axial t hr u st (kN)

Japikse Larjola w/o seal

Larjola w seal Experiment CFD w/o seal CFD w seal (a)

1 2 3 4 5 6 7 8 9

Point number –100

–80 –60 –40 –20 0 20 40 60 80 100

Rela ti ve diff er ence in axial t h ru st (%)

Japikse Larjola w/o seal

Larjola w seal CFD w/o seal CFD w seal

(b)

Figure 7: (a) The axial thrust predicted with three methods, numerically modeled and measured. (b) Relative difference between the

predicted and measured axial thrust.

Nguyen-Schäfer’s method in Figure 12, the method underes- timates the axial thrust because of the overestimated pressure distribution on the shroud side (Figure 10).

Japikse’s method underestimates the pressure distribu- tion on the cavity side, especially when the labyrinth seal is not modeled (Figure 11). The underestimation of the pres- sure force on the back disk results in the underestimated net axial thrust at all studied points.

Larjola’s method estimates the pressure distribution on the shroud side similarly to the numerical model and the methods proposed by Nguyen-Schäfer and Japikse. The axial labyrinth seal is assumed under Larjola’s method, but the radial labyrinth seal is used in the studied compressor.

Despite the radial labyrinth seal, Larjola ’s method gives the closest estimation of the axial thrust value when compared to the measurement in point 9, even though the pressure

1 2 3 4 5 6 7 8 9

Point number 0.0

2.0 4.0 6.0 8.0 10.0 12.0

Axial t hr u st (kN)

Japikse Larjola w/o seal

Larjola w seal CFD w/o seal CFD w seal

(a)

1 2 3 4 5 6 7 8 9

Point number 0.0

2.0 4.0 6.0 8.0 10.0 12.0

Axia l t hr u st (kN)

Japikse Larjola w/o seal

Larjola w seal CFD w/o seal CFD w seal

(b)

1 2 3 4 5 6 7 8 9

Point number 0.0

0.1 0.2 0.3 0.4

Axia l t hr u st (kN)

Japikse Larjola w/o seal

Larjola w seal CFD w/o seal CFD w seal

(c)

Figure 8: (a) Pressure force acting on the shroud side, (b) pressure force acting on the back disk side, and (c) impulse force estimated

numerically and with the three methods, with and without leakage/labyrinth seal.

distribution on the back disk side does not follow a numerical result.

If the measured impeller outlet pressure and ambient pressure are used to estimate the pressure distribution according to Kearton’s [9] theory, the pressure distribution (green dashed line in Figure 12) deviates signi ficantly from the numerical results. Also, the leakage mass flow rate (79–

159 g/s) is more than ten times larger than the numerical result (5 –10 g/s). However, the theoretical pressure distribu- tion (green dotted line) equals the numerical result if the numerical prediction for the seal inlet and outlet pressures is used. The close match between the theoretical and numer- ical pressure distributions is an encouraging result when there are no pressure measurements at the seal inlet and out- let. However, in order to keep the method simple, the CFD results cannot be included in the axial thrust estimation

method, because otherwise the CFD modeling would be required prior to the axial thrust estimation.

The pressure distributions in Figure 12 indicate that close to the design point (point 9), the assumption of the pressure reduction to ambient pressure at the end of the labyrinth seal (at the radius of 62.5 mm) cancels out the effect of the over- estimated pressure in the labyrinth seal on the pressure force.

Hence, Larjola ’s method gives an estimation of the net axial thrust that is in good agreement with the measured one.

However, at other points, the assumption of the ambient pressure between the shaft and the end of the labyrinth seal results in an underestimated pressure force on the back disk side and net axial thrust (point 1 shown in Figure 12).

The main drawbacks of the available methods are that they tend to underestimate the maximum axial thrust, and they require data that are not available at the preliminary

135

67

20

135

63

16 0

0 50 100 150 200 200 150 100 50 0

0

Pressure (kPa) Pressure (kPa)

R adi us (mm) R adi us (mm)

Japikse Larjola w/o seal

Figure 9: Pressure distributions on the shroud and back disk estimated with the three methods at the design point (point 8).

80 300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 250

200 150 100 50 0

Japikse Larjola CFD Experiments

(a)

Japikse Larjola CFD Experiments 80

300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 250

200 150 100 50 0

(b)

Figure 10: Pressure distribution on the shroud. Solid lines: three methods, dotted line: CFD, gray area: discretization error, and circles:

experiments. (a) Point 1. (b) Point 9. Points 2 –8 fall within these two extremes.

design of a turbomachine. The accurate prediction of the pressure distribution on the back disk side causes the most di fficulties in the axial thrust estimation. The conclusion, based on the results presented above, is that Larjola ’s method works best at the point of maximum axial thrust (point 9), but it fails to predict the axial thrust at low rotational speeds.

Since bearing design is aimed at estimating the maximum axial thrust, the accurate prediction of the axial thrust at the point of high rotational speeds and a low flow rate is more

important than it is at lower rotational speeds, where the axial thrust values are lower.

To avoid the underestimation of the axial thrust, which leads to large safety margins and oversized bearings, a more accurate axial thrust estimation method is required. Thus, a new hybrid method is proposed for the estimation of the maximum axial thrust in the preliminary design phase of a radial turbomachine. The hybrid method is presented in the following section.

80 300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 250

200 150 100 50 0

Japikse Larjola w/o seal CFD w/o seal Experiments

(a)

Japikse Larjola w/o seal CFD w/o seal Experiments 80

300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 250

200 150 100 50 0

(b)

Figure 11: Pressure distribution on the back disk without labyrinth seal. Solid lines: three methods, dotted line: CFD, gray area: discretization error, and circles: experiments. (a) Point 1. (b) Point 9. Points 2 –8 fall within these two extremes.

80 300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 250

200 150 100 50 0

Japikse Larjola w seal CFD w seal Experiments

Kearton based on experiments Kearton based on CFD

(a)

Japikse Larjola w seal CFD w seal Experiments

Kearton based on experiments Kearton based on CFD 80

300

100 120 140 160 180 Pressure (kPa)

R adi us (mm)

200 220 240 250

200 150 100 50 0

(b)

Figure 12: Pressure distribution on the back disk with labyrinth seal. Solid lines: three methods, dotted line: CFD, gray area: discretization

error, circles: experiments, and green lines: theory of Kearton. (a) Point 1. (b) Point 9. Points 2 –8 fall within these two extremes.

6. Hybrid Method

The hybrid method combines the best elements of the analyt- ical methods compared in the preceding section. It calculates the net axial thrust as the sum of the impulse force and the forces acting at the impeller eye and nose, at the shroud, and on the back disk side as follows:

F = F eye+nose + F shroud + F impulse − F back disk : ð43Þ The force at the impeller eye and nose is calculated as fol- lows:

F eye+nose = p _inlet A inlet ð compressor/pump Þ, ð44Þ F eye+nose = p outlet A outlet ð turbine Þ: ð45Þ The pressure distribution at the shroud is estimated with a relationship between velocity, pressure and total relative pressure, as suggested by Japikse [4].

p _t = p _inlet + w ² _inlet 2 − U ² _inlet

2

 

ρ _inlet = p _outlet + w ² _outlet 2 − U ² _outlet

2

 

ρ _outlet , ð46Þ where the relative velocity is assumed to be half of the local rotor circumferential velocity.

w = f U, ð47Þ

where f = 0:5. The force at the shroud is calculated for a cen- trifugal compressor/pump and radial turbine as follows:

F shroud = 1 2

ρ 2  1 − f ² 

2πn

ð Þ ²  r ² ₂ + r ² _1,s 



+p _inlet − ρ _inlet 2  1 − f ² 

U ² _inlet i

· π r  ² 2 − r ² _1,s 

compressor/pump

ð Þ,

ð48Þ

F

shroud

= 1 2

ρ 2  1 − f

²



2 πn

ð Þ

²

 r

²1

+ r

²_2,s





+p

outlet

− ρ

outlet

2  1 − f

²

 U

²outlet

i

· π r 

²₁

− r

²_2,s

 turbine

ð Þ,

ð49Þ where ρ is the average gas density between the rotor inlet and outlet.

The impulse force is calculated as follows:

F impulse = q ² _m ρ inlet A inlet

compressor/pump

ð Þ, ð50Þ

F impulse = q ² _m ρ outlet A outlet

turbine

ð Þ: ð51Þ

On the back disk side, the fluid element is assumed to be in radial equilibrium so that the pressure forces balance the centrifugal forces, as suggested by Larjola [5].

dp = ρ f ω ð Þ ² rdr, ð52Þ Table 9: Parameters required for the hybrid method.

Centrifugal compressor/pump Radial in flow turbine

Pressure at the impeller inlet p ₁ Pa p ₁ Pressure at the rotor inlet

Temperature at the impeller inlet T ₁ K T ₁ Temperature at the rotor inlet

Pressure at the impeller outlet p ₂ Pa p ₂ Pressure at the rotor outlet

Temperature at the impeller outlet T ₂ K T ₂ Temperature at the rotor outlet

Ambient pressure p _amb Pa p _amb Ambient pressure

Impeller inlet hub radius r _1,h m r _2,h Rotor outlet hub radius

Impeller inlet shroud radius r _1,s m r _2,s Rotor outlet shroud radius

Impeller outlet radius r 2 m r 1 Rotor inlet radius

Labyrinth seal (outer) radius r _L m r _L Labyrinth seal (outer) radius

Shaft radius r 0 m r 0 Shaft radius

Mass flow rate q _m kg/s q _m Mass flow rate

Rotational speed n rpm n Rotational speed

Fraction, shroud and back disk f 0.5 f Fraction, shroud and back disk

Table 10: Parameters of the centrifugal compressor used in the hybrid method.

p ₁ T ₁ p ₂ T ₂ p _amb r _1,h r _1,s r ₂ r _L r ₀ q _m n

kPa K kPa K kPa mm mm mm mm mm kg/s rpm

Predesign 93.6 283 198.9 357 100.3 20.3 67.5 135.5 96.0 16.0 1.57 27,660

Experimental 94.5 297 199.4 412 100.2 20.3 67.5 135.5 96.0 16.0 1.57 27,725

where f = 0:5 as the absolute tangential velocity is assumed to be half of the local rotor angular velocity. If an axial labyrinth seal is used on the shaft, the pressure varies radially from the rotor outlet to the location of the seal. If a radial labyrinth seal is used, the pressure varies radially from the rotor outlet to the outer radius of the seal, and the pressure varies linearly from the outer radius of the seal to the shaft. For compress- ible flow, the isentropic flow equation gives the relationship between density and pressure as follows:

ρ ρ 2

= p

p 2

  1/γ

: ð53Þ

For incompressible flow, the density is constant. The pressure distribution is derived from (52) for compressible and incompressible flows as follows:

p = p ₂ γ − 1 2 γp 2

ρ 2 f ² ω ² r ² − r ² 2

 

+ 1

 _{ð γ/γ−1 Þ}

compressor

ð Þ,

ð54Þ

p = p 2 + 1

2 ρf ² ω ² r ² − r ² 2

 

pump, incompressible

ð Þ, ð55Þ

p = p ₁ γ − 1 2 γp 1

ρ ₁ f ² ω ² r ² − r ² 1

 

+ 1

 _γ/γ−1

turbine

ð Þ: ð56Þ

The force acting on the back disk side is calculated for a centrifugal compressor/pump and a radial turbine as follows:

F back disk = p _amb πr ² 0 + p _amb + p _r

_L

2

 

π r ² _L − r ² 0

 

+ ð _r

₂

r

_L

p

· 2πrdr compressor/pump ð Þ

ð57Þ

F back disk = p _amb πr ² 0 + p amb + p _r

_L

2

 

π r ² _L − r ² 0

 

+ ð _r

₁

r

L

p · 2πrdr turbine ð Þ, ð58Þ where r _L is either the location of the axial labyrinth seal or the outer radius of the radial labyrinth seal. If there is no lab- yrinth seal in the turbomachine, then the second term in (57) and (58) is neglected and the integral starts from r 0

instead of r _L .

The hybrid method only requires data that are available at the preliminary design of a turbomachine. The required parameters are shown in Table 9.

7. Validation Cases

7.1. Centrifugal Compressor. The centrifugal compressor designed and measured at LUT and presented in Section 2 is used as a validation case. The maximum measured axial thrust is 2190 N, and the measurement uncertainty of axial thrust is 46.3 N. Both the predesign parameters and the experimental results of the point of maximum axial thrust are used in the hybrid method; these are shown in Table 10.

7.2. Radial Pump. The radial pump [17, 18] had 2- dimensional impeller blades, a shrouded impeller, and a seal on the shroud side. According to the drawings [17, 18], there is no labyrinth seal in the back disk cavity. In this paper, the axial thrust is estimated at the design point of the pump since all the required data are available at that point. At the design point, the measured total pressure rise in the impeller is 23.3 kPa and the measured impeller outlet velocity is 3.0 m/s [17]. The incompressible flow and water density of 1000 kg/m ³ are assumed when estimating the static pressure of 120.2 kPa at the impeller outlet. At the design point, the measured axial thrust is 428 N and the measurement uncer- tainty of the axial thrust is 2.7 N [18]. The parameters of the radial pump used in the hybrid method are shown in Table 11.

7.3. Gas Turbines. The results presented in Section 5 show that the analytical methods have difficulties in predicting the axial thrust of a single turbomachine. The task is even more difficult when the centrifugal compressor and the radial turbine are attached to the same shaft. To show the superior- ity of the hybrid method, the predesign parameters of two gas turbines from Aurelia Turbines Oy, Finland, are used in the hybrid method to estimate the net axial thrust. The predesign parameters are proprietary information and are thus not pre- sented in this paper. The gas turbines are low-pressure and high-pressure turbines providing 400 kWe in total. They are later referred to as gas turbines 1 and 2, respectively. The pre- dicted net axial thrust of the gas turbines is compared to the experimental results from Aurelia Turbines Oy. In this paper, the axial thrust is estimated at the design point of the compressors and turbines since the experimental data are available at that point.

8. Validation Results

The new hybrid method is compared to the other analytical methods using the four validation cases. The results of the comparison are shown in Figure 13(a). The three previously published methods predict that the axial thrust is tens of times higher or lower than the measured thrust. In Figure 13(b), only the results of the hybrid method are shown when the preliminary design values (gray color) and experi- mental data (purple color) are used as input data. The maxi- mum relative difference between the measurement and the prediction of the hybrid method of 13% is achieved with the high-pressure gas turbine. For the centrifugal compres- sor, radial pump, and low-pressure gas turbine, the relative di fference between the predicted and measured thrust is less than 10%.

The sensitivity of the input data parameters of the hybrid method on the predicted pressure distributions on the Table 11: Parameters of the radial pump used in the hybrid method.

p ₁ p ₂ p _amb r _1,s r ₂ r ₀ q _m n

kPa kPa kPa mm mm mm kg/s rpm

101.3 120.2 101.3 50.8 101.6 38.1 6.3 620

shroud side (48 and 49) and on the back disk side (54, 55, and 56) was analyzed similarly as with the measurement uncer- tainty, i.e., by calculating the partial derivatives.

Figure 14(a) presents the effect of the parameters in (48 and 49) on the pressure distribution on the shroud side.

Figure 14(b) presents the effect of the parameters in (54, 55, and 56) on the pressure distribution on the back disk side.

The e ffect on the pressure distribution is more equally divided between di fferent parameters on the shroud side than on the back disk side. On the back disk side, the impeller out- let pressure (rotor inlet pressure in the case of the turbine) affects mostly the pressure distribution. The fractions f _s

and f bd on the shroud and back disk sides affect the pressure distributions by only 5 and 6%, respectively.

The fractions f _s and f _bd on the shroud and back disk sides have only a marginal impact on the pressure distribu- tions, as shown in Figure 14. Generally, the average circum- ferential velocity in the back disk cavity is half the impeller ’s local circumferential velocity (f = 0:5) [3–5]. The sensitivity of the net axial thrust on the fraction on the shroud side is analyzed in Figure 15, whereby the fraction on the back disk side remains constant while the fraction on the shroud side varies. The results in Figure 15 indicate that the fraction of 0.5 on the shroud side gives the best

Centrifugal Compressor

Radial Pump

Gas Turbine1

Gas Turbine2 –100

–50 0 50 100 150 200

Rela ti ve diff er ence b etw ee n p redic te d an d me as ur ed net axial t hr u st (%)

Japikse

Larjola w/o seal Larjola w seal Hybrid method (a)

Centrifugal Compressor

Radial Pump

Gas Turbine1

Gas Turbine2 –15

–10 –5 0 5 10 15

Rela ti ve diff er ence b etw ee n p redic te d an d me as ur ed net axial t hr u st (%)

Hybrid method, with pre-design values Hybrid method, with experimental values

(b)

Figure 13: (a) The axial thrust predicted with three previously published methods and the new hybrid method. (b) The axial thrust predicted

with the new hybrid method when using the predesign and experimental values as input data.

prediction of the net axial thrust in the case of the centrifugal compressor and the gas turbines. In the case of the radial pump, the fraction of 0.7 gives a better prediction than the fraction of 0.5.

9. Conclusions

This study compared three analytical axial thrust estimation methods to the numerical and experimental results to ana- lyze their accuracy. According to the results, Japikse’s

method gave the most accurate estimation for the pressure distribution at the shroud and Larjola’s method gave the most accurate estimation for the pressure distribution on the back disk side from the outlet of the impeller of the cen- trifugal compressor to the outer radius of the radial labyrinth seal. Compared to the methods proposed by Japikse and Nguyen-Schäfer, the prime advantage of Larjola’s method was that the pressure distribution was not approximated to change linearly in the back disk cavity; rather, it was esti- mated to vary as a function of radius.

p 1 (24%)

r 2 (17%)

𝜔 (23%)

f _s (5%) r 1,s (6%) U _1,t (8%)

𝜌 ave (11%) 𝜌 1 (6%)

(a)

p 2 (68%)

𝜔 (6%) f bd (6%)

r 2 (11%) r (4%)

𝜌 2 (3%) 𝛾 (1%)

(b)

Figure 14: (a) The effect of the parameters on the pressure distribution on the shroud side when predicted by using the new hybrid method.

(b) The e ffect of the parameters on the pressure distribution on the back disk side when predicted by using the new hybrid method.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fraction f on the shroud side [–]

–150 –100 –50 0 50 100 150 200

Rela ti ve diff er ence b etw ee n p redic te d an d me as ur ed net axial t hr u st (%)

Centrifugal Compressor Radial Pump

Gas Turbine 1 Gas Turbine 2

Fraction f on the back disk side is 0.5.

Figure 15: The sensitivity of the net axial thrust on the fraction on the shroud side when predicted by using the new hybrid method.